Graph/parametric descriptions - Physics C: Mechanics AP Study Notes

Think of kinematics (the study of motion) like telling a story about a moving object. We need different ways to tell that story.

• Graphical Descriptions: Imagine drawing a picture of the story. This is when we use graphs (like a line drawing on a grid) to show how an object's position, velocity (how fast it's going and in what direction), or acceleration (how quickly its velocity is changing) changes over time. It's like a visual diary of its journey.

  • Position-time graph: Shows where an object is at each moment. If you're walking, this graph shows your location on the street at every second.
  • Velocity-time graph: Shows how fast an object is going (and in what direction) at each moment. If you're driving, this graph shows your speedometer reading over time.
  • Acceleration-time graph: Shows how quickly an object's velocity is changing at each moment. If you hit the gas or brake, this graph shows how hard you're doing it.

• Parametric Descriptions: This is like writing a secret code for the story. Instead of one big equation, we use separate equations for each direction (like x, y, and sometimes z) to describe an object's position, velocity, or acceleration. Each equation depends on a single 'master' variable, usually time (t). It's like having separate instructions for moving left/right and moving up/down, both tied to the clock.

  • Imagine a video game character jumping and moving forward at the same time. Its forward motion (x-direction) and its up-and-down motion (y-direction) are described by separate equations, both using the same time 't' to keep them in sync.

Let's track a soccer ball kicked into the air. We want to know exactly where it is at any moment.

1. Graphical Description (Mental Picture): If you could draw its path, it would be a curve (a parabola). If you plotted its height over time, it would go up and then down. If you plotted its horizontal speed over time, it would likely be a straight horizontal line (assuming no air resistance).

2. Parametric Description (The Math Behind the Picture): To get super precise, we'd use two separate equations, both depending on time (t):

   • Horizontal Position (x): Let's say the ball moves horizontally at a steady 10 meters per second. Its horizontal position at any time 't' would be x(t) = 10 * t. (This means if 1 second passes, it's 10m away; if 2 seconds pass, it's 20m away, and so on).
   • Vertical Position (y): This one is trickier because gravity pulls it down. Let's say it starts at 0 height, is kicked upwards at 15 meters per second, and gravity pulls it down at 9.8 meters per second squared (a measure of how much gravity changes its speed). Its vertical position at any time 't' would be y(t) = 15*t - 0.5 * 9.8 * t^2. (The 15*t is its initial upward push, and the -0.5 * 9.8 * t^2 is how much gravity pulls it down over time).

   By using these two equations together, we can plug in any time 't' and get the ball's exact (x, y) coordinates. It's like having a GPS for the soccer ball!